Quantum dynamics of non-rigid systems comprising two polyatomic fragments

MOLECULARPHYSICS, 1983, VOL. 50, No. 5, 1025-1043

Quantum dynamics of non-rigid systems comprising two polyatomic fragments by G. BROCKS and A. VAN DER AVOIRD Institute of Theoretical Chemistry, University of Nijmegen, Toernooiveld, Nijmegen, The Netherlands B. T. S U T C L I F F E Chemistry Department, University of York, Heslington, York Y01 5DD, England and J. T E N N Y S O N SERC Daresbury Laboratory, Daresbury, Warrington WA4 4AD, Cheshire, England

(Received 26 May 1983 ; accepted 4 ffuly 1983) We combine earlier treatments for the embedding of body-fixedcoordinates in linear molecules with the close-coupling formalism developed for atomdiatom scattering and derive a hamiltonian which is most convenient for describing the nuclear motions in van der Waals complexes and other non-rigid systems comprising two polyatomic fragments, .4 and B. This hamiltonian can still be partitioned in the form I~A + I2IB+ I~INT, just as the space-fixed hamiltonian. The body-fixed form, however, has several advantages. We discuss solution strategies for the rovibrational problem in non-rigid dimers, based on this partitioning of the hamiltonian. Finally, in view of the size of the general polyatomic-polyatomic case, we suggest problems which should be currently practicable.

1. INTRODUCTION The search for suitable coordinate systems and hamiltonians for the polyatomic vibration-rotation problem has excited much scientific interest over the years [1-12]. For near-rigid systems, with localized or small amplitude vibrations, the use of Eckart coordinates has proved fruitful [1-5, 13, 14]. However, for one or more large amplitude internal motions the simple Eckart hamiltonian is unsatisfactory. This feature was recognized very early by Sayvetz [2] who modified the original approach of Eckart [1] to try and deal with the difficulties inherent in treating large amplitude internal motions. The problems arise because the transformation that leads from the hamiltonian in the laboratory-fixed coordinate system to the internal hamiltonian, which is expressed in some suitably chosen body-fixed coordinate system, has a jacobian which is singular for some values of the coordinates. Thus the transformed internal hamiltonian is not everywhere well defined and, consequently, has a domain that is more restricted than all square-intcgrablc functions of the body-fixed coordinates. The difficulties associated with this restriction of the domain usually manifest themselves in divergent expectation values of the internal hamiltonian between seemingly reasonable functions. Thus it is well known

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G. Brocks et al.

[15-18] that if an attempt is made to treat a triatomic that has a large amplitude bending mode, in the Eckart approach, then divergence can occur in expectation values of functions that allow the system to become linear. This sort of problem can be solved by using the Sayvetz modification of the Eckart approach and such a way out has been attempted [19-22]. It can also be solved by abandoning the Eckart approach altogether, based as it is on the idea of an equilibrium geometry for the system, and various different formulations have been given [6-10, 23-26]. However, because of the complexity of the problem, use of these hamiltonians has been restricted to triatomic systems and a few symmetric tetra-atomic (AB)~ van der Waals complexes [6, 25-27]. With the exception of a recent calculation on H e H F [28], all these calculations have included some simplifying approximation involving the decoupling or freezing of certain vibrational modes. A popular approach, applied to several van der Waals complexes [6, 7, 10, 23-28] as well as H20 [29], K C N [10, 16], LiCN [30] and CH2 + [31], has been to work in so-called (di)atom-diatom collision coordinates. These are defined as the distance between the monomer centres of mass, the angles describing the orientations of the monomers, the internal monomer coordinates and the overall rotation angles. Several hamiltonians have been used to obtain the bound rovibrational states of systems treated as collision complexes using approaches which show strong analogies to atom- [32], diatom- [33] and electron- [34] diatom scattering problems. The main difference between the various hamiltonians is the embedding of the coordinate system. For atom-diatom systems alone, space fixed coordinates [7] and at least three different embeddings of body-fixed coordinates [8, 10, 30] have been used. In recent work we have favoured the use of coordinates which have R, the interaction or collision coordinate, embedded along the z-axis. This system has computational advantages over space-fixed coordinates and allows the simplifying approximation of neglect of off-diagonal Coriolis interactions, which has proved useful in many calculations [10, 26, 30, 35]. This embedding is favoured over one which fixes the axis(es) in one fragment of the complex, el. Istomin et al. [8], as it is easier to generalize. Fixing R along the z-axis is not sufficient fully to body-fix the coordinates as it only defines two of three possible embedding Euler angles. A fully embedded hamiltonian for the atom-diatom problem was recently given by Tennyson and Sutcliffe [10], but as we show (in Appendix A) this is not convenient in larger systems. However, as shown by Tennyson and Sutcliffe, there is a strong equivalence between the matrix problems generated by the two embeddings. In this paper, we present a general hamiltonian for the rovibrational (and scattering) problem of a system A B , where A and B are polyatomic molecules. This hamiltonian can be expressed in terms of fragment hamiltonians and an interaction hamiltonian /t =/}a + ~B +/linT (1) and it is derived in such a manner t h a t / t A and /QB are the rovibrational hamiltonians of the isolated fragments for which conventional Eckart hamiltonians [1-5] are appropriate. Our hamiltonian is suitable for any system which contains two (near) rigid fragments undergoing large amplitude motions. Thus it is particularly convenient for van der Waals dimers.

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Q u a n t u m dynamics o[ v a n der W a a l s dimers

By embedding the hamiltonian in this manner, we follow several earlier works on atom-diatom and diatom-diatom systems. Our hamiltonian extemds to larger (polyatomic) molecules, however, and, moreover, we hope to clarify the situation with regard to the form and commutation relationships of the resulting pseudo-angular momentum operators (§2). In §§3 and 4 we discuss convenient and practical solution strategies for the problem. An approach to the scattering of two polyatomic molecules which is similar to the present one has been proposed in 1953 by Curtiss [36], who based himself on the pioneering work by Hirschfelder et al. [37, 38]. Curtiss starts out in bodyfixed coordinates (defined by three embedding angles). However, by rotating and recoupling his basis functions he actually derives the close-coupled equations in space-fixed coordinates.

2. HAMILTONIAN Within the Born-Oppenheimer approximation, the hamiltonian for nuclear motion is

H=

h 2

i=1

1 Vi2(xi ) + V,

(2)

where Vi2(xi) is the laplacian for the ith nucleus and V the potential. If the N-nucleus system is divided into fragment A with N a nuclei and fragment B with N B nuclei, then the space (or laboratory) fixed coordinates can be ordered xi, i = l , 2

. N. ~ ,.N A.+ I , 1, ...,

..

,i. }

(3)

N B.

The overall translational motion is removed and the relative translation of the two monomers A and B is defined by the following transformation [36] ti a = x i - XA, tiB=XNA+it o =

X A

-

XB,

i = 1 ..., N a - 1 ] i = 1 ..., N B - - 1

(4)

XB,

l

X = M - I ( M a X a + M B XB) , with

XA=MA -1 E mixi,

i=1 N,i N X B -~-MB -1 E i=N,~+1mixi,

M a = ~" m i,

i=1 N.4 1 N MB = ~ i=NA+Imi'

(5)

M = M a + MB,

where x i denotes a vector of coordinates in the original frame and t i in the new frame.

G. Brocks et al.

1028 Use of the chain rule gives

N 1 E - - V~(xi) = M - t V~(X) +/~-1 V~(to) i=l ms NF-I

+ E

E

Gii F V(tiF). V(tiF),

(6)

i,j=l

F= A, B

where the terms in ti F vanish if NF < 2 and

(F=A,B),}

G i j F = ~iimg -1 -- M F -1,

(7)

t~-1 = MA -1 + MB -1. The first term in (6), M -1 V2(X), corresponds to the free translation of the centre of mass and can be separated off. The remaining terms form the kinetic energy operator of the rovibrational hamiltonian of the A B system, expressed in a coordinate frame which is body-fixed with respect to translations, but still parallel to the laboratory frame. This frame is usually called space-fixed. If the separation of the monomer centres of mass is not done for all particles in one step, as in (4), but pairwise in (NF-- 1) steps for each monomer F, which defines the so-called Jacobi coordinates ti, then the matrix G becomes diagonal

Gij F = ~,~jl~i-1,

(F = .4, B),

(7')

where/~i is the mass of the ith reduced particle. Next, we wish to separate off the overall rotations of the system by defining a body-fixed coordinate frame. The question is how to fix this frame on a nonrigid system, where the equilibrium structure, which could be used to define the Eckart embedding conditions [1], may not be meaningful. It seems natural to single out the vector t 0= R which connects the centres of mass of the two fragments .4 and B and to embed the body-fixed frame with the z-axis along R (following the Pack and Hirschfelder [34] treatment of diatomic systems). If R has the polar angles (fi, a) with respect to the space-fixed system, this embedding is achieved by the orthogonal transformation

z0 = ¢ ~ to =

(i)

zi F = C r ti E,

,

(F = A, B),

(8 a)

(8 b)

with the matrix E, that corresponds to a rotation over two Euler angles, a and fl, given by /cosficosa -sina sinficosa\ C =/[cos fi sin ~ \

-sinfl

cos ~ 0

sin fi sin a ) .

(9)

cos]3

It is possible to define a third embedding angle Y but, as we show in Appendix A, this leads to a less convenient form of the hamiltonian.

Quantum dynamics o[ van der Waals dimers

1029

We have to consider the effect of the rotation ¢ on the hamiltonian, in particular on the kinetic energy operator. In order to express the differential operators of the old coordinates (4) in the new ones, we have to use the chain rule. In applying this rule, however, it must be remembered that the embedding condition (8) makes each of the new coordinates z f a function of t 0, because the matrix C depends on the angles a and /3 which occur in t o = R = (R sin/3 cos ~, R sin 13 sin ~, R cos/3). T h u s

O OR O 0~ O 0/3 b N,-, Ozi,~F O ato,~- Oto,~ OR l- a-~o,~~-I---Oto,,O/3 -- + ~ i~=, ~ Oto., Ozi,, F

(10)

a n d we can substitute

OR Oto,~ C~z, 0~

(11)

Oto,----~= (R sin/3)-' C~u,

o/3

-R

-1

C~x

ato,~ and

Ozi,yF OC~y • _ OC~ Oto.~ = ~-~o, ti,," = g~"-~oj Cg,zi.pF,

(12)

with --

0to, ~

aa 0to,g

-F-

(13)

0/3 ato, g"

The results of these substitutions can be written more simply in terms of the following angular m o m e n t u m operators and their transformation properties (~a~,v is the Levi-Civita antisymmetric tensor ; A,/~, v = x, y or z)

~',~"Ea.vti.#F Oti,O

h ;a(ti F) =~-

(14 a)

vF ,

N~-t

iF=

Z l(tiF),

(14 b)

i=l

i = E ir = i ~ + i ~,

(14 c)

F

h 0 h~ la = ~" ~ Ea""t°'" ~t o ~ ,

R

0 ~R~

(14 d)

and the total angular m o m e n t u m J

=l+j=l-l--ja+j B.

(14e)

The transformation of the operators j(tiF), jR and j is easy because C is not a function of the coordinates ti F and we can write, using (8) 0

OZi, y F

0

Otij F= ~ Oti,gF ~Zi,~ F a = Y~ C~. ,7

OZi,~IF"

(15)

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G. Brocks

et al.

Using this result and the transformation properties of the Levi-Civita tensor, this yields ----

E

14"

h ~ Ca~~,~ ~ ~' =-'z , E~Zi'~F~Zi,

(16)

which shows that the components of j transform as the components of a vector j(tiF ) = Ci(ziF), with the same expressions (14

a-c) for

(17)

| holding in the body-fixed coordinates

Zi F .

The transformation of |, equation (14 d), is more complicated because one must use the chain rule expressions (10) to (13). First, we substitute (11 ) into (13) and rewrite (10) as e

~to, ~

=R_IC~ x

~_~jy +R_~C~" -~fl

cosecfl~_~a_cotfl~]~+~ L + C~, ~-~,

(18)

with the angular momentum operator j = j A + j B expressed in body-fixed coordinates ,i F. Then, using (8 a), it is straightforward to derive the angular momentum operator |, (14 d), also in body-fixed coordinates. Instead, we write the result for the total angular momentum ) = | + j, which is slightly simpler. Just as j, see (17), 3 transforms as 3(ti) = C3(zi),

(19)

with the components of 3 in the body-fixed coordinates J~ = - _ cosec •

z

~

+ cot

jr= h ~ i @'

Zi F, O~and

fl, given by

flj~, j

(20)

J =L. The form of this operator is unusual, it is different from the more familiar expression for J in a body-fixed frame [34] which is embedded by three Euler angles, ~, fl and 7, rather than just two. We observe that its components do not even satisfy angular momentum commutation relations

Ida, do] = [dy, d,] = 0,

]

[Jx,Juj=hcotflfix+ ] , l

(21)

Quantum dynamics o[ van der Waals dimers

1031

so that we must call d a pseudo-angular momentum operator. ponents of 3 do not commute with j [']x, ]a] = - - cot

Also, the com-

fl%aj~,

[ J u ' £ ] =0, h - 7 e~aaJa

with A, ~ = x, y, z.

j

(22)

Watson [5], in his isomorphic hamiltonian for linear molecules, restores the usual body-fixed expressions for d by introducing an artificial third rotation angle y. In the next section we show that the action of our d on a suitably chosen basis is actually quite simple, however, so that there is no need to invoke this extraneous angle. In Appendix A we demonstrate that embedding with three (physically defined, rather than extraneous) Euler angles only leads to a formalism which is less transparent and more difficult to apply. After all this preliminary work it is not too tedious to write the kinetic energy operator (6) in body-fixed coordinates, =, fl, R, l i r "~ we only have to substitute (15) for the derivatives V(tiF) and (18) for V(t0). The derivatives V(tiF) commute with the matrix C and so we easily find (using C T C = 1) that the terms h 2

Nv-~

----2 i,i~=l GijF V(zi F) V(zjF), /~F =

( F = A , B)

(23)

are form-invariant. If NF